where \(A_j^\dagger \) , \(A_k\)  are the fermionic creation and annihilation operators, \(N_l\)  are the occupation-number operators, \(\tau \) , \(\nu \)  are coefficient matrices, and the summation is over \(n\)  spin orbitals. The specific definitions of these fermionic operators are given in [sec:prelimquantization]Section sec:prelimquantization. We say the interactions are \(d\) -sparse if there are at most \(d\)  nonzero elements within each row/column of \(\tau \)  and \(\nu \) . This model represents various systems arising in physics and chemistry, including the electronic-structure Hamiltonians in the plane-wave basis [1]} and the Fermi-Hubbard model [2]}, [3]}.
